Laplace's Method Revisited: Weak Convergence of Probability Measures

Abstract

Let $Q$ be a fixed probability on the Borel $\sigma$-field in $R^n$ and $H$ be an energy function continuous in $R^n$. A set $N$ is related to $H$ by $N = \{x \mid\inf_yH(y) = H(x)\}$. Laplace's method, which is interpreted as weak convergence of probabilities, is used to introduce a probability $P$ on $N$. The general properties of $P$ are studied. When $N$ is a union of smooth compact manifolds and $H$ satisfies some smooth conditions, $P$ can be written in terms of the intrinsic measures on the highest dimensional mainfolds in $N$.